tweak whole site: caps for Reader monad, etc

[lambda.git] / week7.mdwn

diff --git a/week7.mdwn b/week7.mdwn
index abb6cfe..961b024 100644 (file)
--- a/week7.mdwn
+++ b/week7.mdwn
@@ -164,7 +164,7 @@ theory of accommodation, and a theory of the situations in which
 material within the sentence can satisfy presuppositions for other
 material that otherwise would trigger a presupposition violation; but,
 not surprisingly, these refinements will require some more
 material within the sentence can satisfy presuppositions for other
 material that otherwise would trigger a presupposition violation; but,
 not surprisingly, these refinements will require some more

-sophisticated techniques than the super-simple option monad.]
+sophisticated techniques than the super-simple Option monad.]

 
 
 Monads in General
 
 
 Monads in General


@@ -227,7 +227,7 @@ that provides at least the following three elements:
 
        So `unit` is a way to put something inside of a monadic box. It's crucial
        to the usefulness of monads that there will be monadic boxes that
 
        So `unit` is a way to put something inside of a monadic box. It's crucial
        to the usefulness of monads that there will be monadic boxes that

-       aren't the result of that operation. In the option/maybe monad, for
+       aren't the result of that operation. In the Option/Maybe monad, for

        instance, there's also the empty box `None`. In another (whimsical)
        example, you might have, in addition to boxes merely containing integers,
        special boxes that contain integers and also sing a song when they're opened. 
        instance, there's also the empty box `None`. In another (whimsical)
        example, you might have, in addition to boxes merely containing integers,
        special boxes that contain integers and also sing a song when they're opened. 


@@ -238,7 +238,7 @@ that provides at least the following three elements:
 
 *      Thirdly, an operation that's often called `bind`. As we said before, this is another
        unfortunate name: this operation is only very loosely connected to
 
 *      Thirdly, an operation that's often called `bind`. As we said before, this is another
        unfortunate name: this operation is only very loosely connected to

-       what linguists usually mean by "binding." In our option/maybe monad, the
+       what linguists usually mean by "binding." In our Option/Maybe monad, the

        bind operation is:
 
                let bind u f = match u with None -> None | Some x -> f x;;
        bind operation is:
 
                let bind u f = match u with None -> None | Some x -> f x;;


@@ -260,7 +260,7 @@ that provides at least the following three elements:
 
        The guts of the definition of the `bind` operation amount to
        specifying how to unbox the monadic value `u`.  In the `bind`
 
        The guts of the definition of the `bind` operation amount to
        specifying how to unbox the monadic value `u`.  In the `bind`

-       operator for the option monad, we unboxed the monadic value by
+       operator for the Option monad, we unboxed the monadic value by

        matching it with the pattern `Some x`---whenever `u`
        happened to be a box containing an integer `x`, this allowed us to
        get our hands on that `x` and feed it to `f`.
        matching it with the pattern `Some x`---whenever `u`
        happened to be a box containing an integer `x`, this allowed us to
        get our hands on that `x` and feed it to `f`.


@@ -281,7 +281,7 @@ that provides at least the following three elements:
        For each new monadic type, this has to be worked out in an
        useful way.
 
        For each new monadic type, this has to be worked out in an
        useful way.
 

-So the "option/maybe monad" consists of the polymorphic `option` type, the
+So the "Option/Maybe monad" consists of the polymorphic `option` type, the

 `unit`/return function, and the `bind` function.
 
 
 `unit`/return function, and the `bind` function.
 
 


@@ -318,7 +318,7 @@ OCaml. In fact, the `<-` symbol already means something different in OCaml,
 having to do with mutable record fields. We'll be discussing mutation someday
 soon.)
 
 having to do with mutable record fields. We'll be discussing mutation someday
 soon.)
 

-As we proceed, we'll be seeing a variety of other monad systems. For example, another monad is the list monad. Here the monadic type is:
+As we proceed, we'll be seeing a variety of other monad systems. For example, another monad is the List monad. Here the monadic type is:

 
        # type 'a list
 
 
        # type 'a list
 


@@ -345,7 +345,7 @@ of `'b list`s into a single `'b list`:
        # List.concat [[1]; [1;2]; [1;3]; [1;2;4]]
        - : int list = [1; 1; 2; 1; 3; 1; 2; 4]
 
        # List.concat [[1]; [1;2]; [1;3]; [1;2;4]]
        - : int list = [1; 1; 2; 1; 3; 1; 2; 4]
 

-So now we've seen two monads: the option/maybe monad, and the list monad. For any
+So now we've seen two monads: the Option/Maybe monad, and the List monad. For any

 monadic system, there has to be a specification of the complex monad type,
 which will be parameterized on some simpler type `'a`, and the `unit`/return
 operation, and the `bind` operation. These will be different for different
 monadic system, there has to be a specification of the complex monad type,
 which will be parameterized on some simpler type `'a`, and the `unit`/return
 operation, and the `bind` operation. These will be different for different


@@ -407,7 +407,7 @@ them from hurting the people that use them or themselves.
        If you don't understand why the lambda form is necessary (the
        "fun x -> ..." part), you need to look again at the type of `bind`.
 
        If you don't understand why the lambda form is necessary (the
        "fun x -> ..." part), you need to look again at the type of `bind`.
 

-       Some examples of associativity in the option monad (bear in
+       Some examples of associativity in the Option monad (bear in

        mind that in the Ocaml implementation of integer division, 2/3
        evaluates to zero, throwing away the remainder):
 
        mind that in the Ocaml implementation of integer division, 2/3
        evaluates to zero, throwing away the remainder):
 


@@ -428,7 +428,7 @@ them from hurting the people that use them or themselves.
 
 Of course, associativity must hold for *arbitrary* functions of
 type `'a -> 'b m`, where `m` is the monad type.  It's easy to
 
 Of course, associativity must hold for *arbitrary* functions of
 type `'a -> 'b m`, where `m` is the monad type.  It's easy to

-convince yourself that the `bind` operation for the option monad
+convince yourself that the `bind` operation for the Option monad

 obeys associativity by dividing the inputs into cases: if `u`
 matches `None`, both computations will result in `None`; if
 `u` matches `Some x`, and `f x` evalutes to `None`, then both
 obeys associativity by dividing the inputs into cases: if `u`
 matches `None`, both computations will result in `None`; if
 `u` matches `Some x`, and `f x` evalutes to `None`, then both


@@ -505,7 +505,7 @@ The `lift` operation we asked you to define for last week's homework is a common
        # let even x = (x mod 2 = 0);;
        val g : int -> bool = <fun>
 
        # let even x = (x mod 2 = 0);;
        val g : int -> bool = <fun>
 

-`even` has the type `int -> bool`. Now what if we want to convert it into an operation on the option/maybe monad?
+`even` has the type `int -> bool`. Now what if we want to convert it into an operation on the Option/Maybe monad?

 
        # let lift g = fun u -> bind u (fun x -> Some (g x));;
        val lift : ('a -> 'b) -> 'a option -> 'b option = <fun>
 
        # let lift g = fun u -> bind u (fun x -> Some (g x));;
        val lift : ('a -> 'b) -> 'a option -> 'b option = <fun>


@@ -518,7 +518,7 @@ also define a lift operation for binary functions:
 
 `lift2 (+)` will now be a function from `int option`s  and `int option`s to `int option`s. This should look familiar to those who did the homework.
 
 
 `lift2 (+)` will now be a function from `int option`s  and `int option`s to `int option`s. This should look familiar to those who did the homework.
 

-The `lift` operation (just `lift`, not `lift2`) is sometimes also called the `map` operation. (In Haskell, they say `fmap` or `<$>`.) And indeed when we're working with the list monad, `lift f` is exactly `List.map f`!
+The `lift` operation (just `lift`, not `lift2`) is sometimes also called the `map` operation. (In Haskell, they say `fmap` or `<$>`.) And indeed when we're working with the List monad, `lift f` is exactly `List.map f`!

 
 Wherever we have a well-defined monad, we can define a lift/map operation for that monad. The examples above used `Some (g x)` and so on; in the general case we'd use `unit (g x)`, using the specific `unit` operation for the monad we're working with.
 
 
 Wherever we have a well-defined monad, we can define a lift/map operation for that monad. The examples above used `Some (g x)` and so on; in the general case we'd use `unit (g x)`, using the specific `unit` operation for the monad we're working with.
 


@@ -545,7 +545,7 @@ and so on. Here are the laws that any `ap` operation can be relied on to satisfy
        ap (unit f) (unit x) = unit (f x)
        ap u (unit x) = ap (unit (fun f -> f x)) u
 
        ap (unit f) (unit x) = unit (f x)
        ap u (unit x) = ap (unit (fun f -> f x)) u
 

-Another general monad operation is called `join`. This is the operation that takes you from an iterated monad to a single monad. Remember when we were explaining the `bind` operation for the list monad, there was a step where
+Another general monad operation is called `join`. This is the operation that takes you from an iterated monad to a single monad. Remember when we were explaining the `bind` operation for the List monad, there was a step where

 we went from:
 
        [[1]; [1;2]; [1;3]; [1;2;4]]
 we went from:
 
        [[1]; [1;2]; [1;3]; [1;2;4]]


@@ -583,7 +583,7 @@ Continuation monad.
 But first, we'll look at several linguistic applications for monads, based
 on what's called the *Reader monad*.
 
 But first, we'll look at several linguistic applications for monads, based
 on what's called the *Reader monad*.
 

-##[[Reader monad for Variable Binding]]##
+##[[Reader Monad for Variable Binding]]##

 
 

-##[[Reader monad for Intensionality]]##
+##[[Reader Monad for Intensionality]]##